Here, they demonstrate the existence of a number that is not the sum or difference of two primes powers; using covering systems.

First, they construct the covering systems and obtain some congruences that demonstrate that the given number is
not the sum or difference of a power of 2 and a prime. This I understand without any difficulty.

Then, for proving that the number such obtained is not the sum of a power of two and a POWER of prime,they introduce some
extra congruences. The basic idea is that if one could show the reqd. number to be multiple of a product of two or more primes,
then it won't be a prime power.

That is fine, but take a look at the primes in extra congruences. From where are these extra primes selected? e.g. on page 2 of the paper, we have: